The torque and thus the horsepower
generated by an induction motor is a function of its slip – the
difference between its rotor speed and the speed of the synchronous
field in the stator – and,

The work extracted from a motor by a
centrifugal machine like a pump is a function of the cube of it’s
speed, then

For a given pump attached to a given
motor, there will only be one speed where the bhp
extracted by the pump will exactly balance the HP generated by the
motor.

My recent insight regarding all of this was that if you obtain
the speed vs. load data for the motor you are considering and then
plot the speed vs. bhp required for the pump you are considering on
the same set of axis, then the point where the two curves cross is
the balance point. In other words, you can develop a graphic
solution to the problem vs. an algebraic solution. The graph below
illustrates this for the three motors and pump associated with the
preceding table.

I like graphic solutions because they let you see your answer in
a broader perspective (plus they may be a bit more user-friendly if
you are math-phobic). If I solved this problem by setting
simultaneous equations, one for the pump power vs. speed and one
for the motor power vs. speed equal to each other a couple of
things come up.

For one thing, I have to develop the equations in the first
place. An exact solution for the motor speed vs. power equation
would involve deriving a polynomial that described the actual
relationship as depicted in the manufacturer’s data. Of course
there are tolerances associated with the manufacturer’s data, so
it’s probably reasonable (but not exact) to assume that the pump
speed to power relationship is linear.

In a similar vein, for the pump, an exact solution would involve
picking flow, head and efficiency points off of the pump
performance curve, calculating brake horsepower from the pump power
equation, and then developing a polynomial that described the line.
Or, I could assume that the pump speed vs. power relationship
follows the affinity laws, a reasonable assumption but not
exact.

Either way, I end up with two equations that contain some
assumptions that can be solved simultaneously for a specific
number. And, I probably had to draw a graph to develop the
polynomial anyway. By simply plotting both curves on the same axis
and looking at where they intersect, I can get the answer I am
looking for with less effort. And, I view it in the context
of the entire power vs. speed spectrum.

In the next post, we’ll look at some resources that will allow
you to get the data you need to do the graphic analysis.